Math Problem Statement

fill out this graph correctly

Fill out this graph correctly

Solution

Let's fill out the missing values in the tables for each equation based on the information provided.

Equation 1: y=23x+3y = \frac{2}{3}x + 3

Given:

  • x=0x = 0, y=3y = 3 (already filled)
  • x=6x = 6, y=7y = 7 (already calculated in the formula box)
  • x=10x = 10, y=12y = 12 (already filled)

To complete:

Let's select an additional xx value, such as x=3x = -3, and calculate yy.

  1. For x=3x = -3: y=23(3)+3=2+3=1y = \frac{2}{3}(-3) + 3 = -2 + 3 = 1 So, when x=3x = -3, y=1y = 1.

  2. Another value, for x=15x = 15: y=23(15)+3=10+3=13y = \frac{2}{3}(15) + 3 = 10 + 3 = 13 So, when x=15x = 15, y=13y = 13.

Now, the completed table for y=23x+3y = \frac{2}{3}x + 3 will be:

xxyy
03
67
1012
-31
1513

Equation 2: y=4x+4y = -4x + 4

Given:

  • x=0x = 0, y=4y = 4 (already filled)
  • x=12x = 12, y=44y = -44 (formula needs to be calculated in the box)
  1. Calculating yy for x=12x = 12: y=4(12)+4=48+4=44y = -4(12) + 4 = -48 + 4 = -44 So, when x=12x = 12, y=44y = -44.

To complete:

Let's choose additional xx values, such as x=1x = -1 and x=5x = 5.

  1. For x=1x = -1: y=4(1)+4=4+4=8y = -4(-1) + 4 = 4 + 4 = 8 So, when x=1x = -1, y=8y = 8.

  2. For x=5x = 5: y=4(5)+4=20+4=16y = -4(5) + 4 = -20 + 4 = -16 So, when x=5x = 5, y=16y = -16.

Now, the completed table for y=4x+4y = -4x + 4 will be:

xxyy
04
12-44
-18
5-16

If you need further details on any steps or explanations, feel free to ask!

Follow-up Questions

  1. How would you determine the slope and intercept from each equation?
  2. What is the significance of positive vs. negative slopes?
  3. How can you verify these points graphically?
  4. How does changing the slope or intercept affect the line's direction and position?
  5. What patterns do you notice in the y-values as x increases or decreases?

Tip: Remember, the slope represents how much yy changes with each increase of 1 in xx.

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Math Problem Analysis

Mathematical Concepts

Algebra
Linear Equations
Slope-Intercept Form

Formulas

y = mx + b

Theorems

Slope-Intercept Form of a Line

Suitable Grade Level

Grades 7-9